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May 2025 Section II

0/54 marks

2025 · 6 questions · 24 parts · 54 marks

0Correct0Partial0Wrong24Unattempted

1(A)1(B)1(C)1(D)2(A)2(B)2(C)2(D)3(A)3(B)3(C)3(D)4(A)4(B)4(C)4(D)5(A)5(B)5(C)5(D)6(A)6(B)6(C)6(D)

Context for question 1

An invasive species of plant appears in a fruit grove at time

t = 0

and begins to spread. The function

C

defined by

C (t) = 7.6 arctan (0.2 t)

models the number of acres in the fruit grove affected by the species

t

weeks after the species appears. It can be shown that

C^{'} (t) = \frac{38}{25 + t ^{2}}

. (Note: Your calculator should be in radian mode.)

1(A)Definite Integrals and the Fundamental TheoremEasy2 marks

Find the average number of acres affected by the invasive species from time

t = 0

to time

t = 4

weeks. Show the setup for your calculations.

1(B)Mean Value Theorem and L'Hopital's RuleEasy2 marks

Find the time

t

when the instantaneous rate of change of

C

equals the average rate of change of

C

over the time interval

0 \leq t \leq 4

. Show the setup for your calculations.

1(C)Limits and ContinuityMedium2 marks

Assume that the invasive species continues to spread according to the given model for all times

t > 0

. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.

1(D)Stationary Points, Concavity and Curve SketchingHard3 marks

At time

t = 4

weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function

A

, defined by

A (t) = C (t) - \int_{4}^{t} 0.1 \cdot ln (x) d x

, models the number of acres affected by the species over the time interval

4 \leq t \leq 36

. At what time

t

, for

4 \leq t \leq 36

, does

A

attain its maximum value? Justify your answer.